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Unformatted text preview: 2 !...nr ! = n
n1 , n2 , . . . , nr Same as number of anagrams of nletter word with n1 repeats of
ﬁrst letter, n2 of second, etc.!
Example: How many ways to break the class (of 33) into 11 groups of
3? Math 30530 (Fall 2012) Counting September 13, 2013 8 / 12 Splitting a set up into classes of given sizes
In how many ways can we split (partition) a set of size n into r parts,
with the ﬁrst part having size n1 , the second size n2 , . . ., the r th size nr
(and n = n1 + . . . + nr )?
r = 2: same as choosing subset of size n1 (what’s left forms
n
second part of size n2 ), so n1
General r : Two expressions
n
n1 n−n1
n2 ... n−n1 −...−nr −1
nr n!
n1 !n2 !...nr ! = n
n1 , n2 , . . . , nr Same as number of anagrams of nletter word with n1 repeats of
ﬁrst letter, n2 of second, etc.!
Example: How many ways to break the class (of 33) into 11 groups of
3?
33
33!
/11! =
≈ 5.99 × 1020
3, 3, . . . , 3
11!3!11
Math 30530 (Fall 2012) Counting September 13, 2013 8 / 12 Selecting k items from n, WITH REPLACEMENT
In how many ways can we pull out k items from among n different,
distinct objects, if each time we pull out an item, we note it and put it
back? Math 30530 (Fall 2012) Counting September 13, 2013 9 / 12 Selecting k items from n, WITH REPLACEMENT
In how many ways can we pull out k items from among n different,
distinct objects, if each time we pull out an item, we note it and put it
back?
Order matters: (we produce a list (ﬁrst item, second, . . ., last))
nk Math 30530 (Fall 2012) Counting September 13, 2013 9 / 12 Selecting k items from n, WITH REPLACEMENT
In how many ways can we pull out k items from among n different,
distinct objects, if each time we pull out an item, we note it and put it
back?
Order matters: (we produce a list (ﬁrst item, second, . . ., last))
nk
Example: 8 digit numbers with prime digits? 48 Math 30530 (Fall 2012) Counting September 13, 2013 9 / 12 Selecting k items from n, WITH REPLACEMENT
In how many ways can we pull out k items from among n different,
distinct objects, if each time we pull out an item, we note it and put it
back?
Order matters: (we produce a list (ﬁrst item, second, . . ., last))
nk
Example:...
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This note was uploaded on 01/31/2014 for the course MATH 30530 taught by Professor Hind,r during the Fall '08 term at Notre Dame.
 Fall '08
 Hind,R
 Counting, Probability

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